Question

1. Calculate the work done when 1,36 kg of gas is compressed isothermally from 138 kPa to 1 380 kPa. Isothermal temperature is 77 °C and R is 0,286 kJ/kg.K.

Answer 1

To calculate the work done in compressing the gas isothermally, we can use the formula for isothermal work done on an ideal gas:

\[
W = m \cdot R \cdot T \cdot \ln\left(\frac{P_2}{P_1}\right)
\]

Where:
- \( W \) = work done
- \( m \) = mass of the gas (in kg)
- \( R \) = specific gas constant (in kJ/kg·K)
- \( T \) = absolute temperature (in K)
- \( P_1 \) = initial pressure (in kPa)
- \( P_2 \) = final pressure (in kPa)
- \( \ln \) = natural logarithm

### Given data:
- \( m = 1.36 \) kg
- \( R = 0.286 \) kJ/kg·K
- \( T = 77 \) °C = \( 77 + 273.15 = 350.15 \) K
- \( P_1 = 138 \) kPa
- \( P_2 = 1380 \) kPa

### Calculating the Work Done:

1. **Calculate the Natural Logarithm of the Pressure Ratio:**
\[
\frac{P_2}{P_1} = \frac{1380 \text{ kPa}}{138 \text{ kPa}} = 10
\]
\[
\ln\left(\frac{P_2}{P_1}\right) = \ln(10) \approx 2.3026
\]

2. **Substitute the Values into the Work Done Formula:**
\[
W = 1.36 \, \text{kg} \cdot 0.286 \, \text{kJ/kg·K} \cdot 350.15 \, \text{K} \cdot 2.3026
\]

3. **Calculate the Result Step-by-Step:**
- First calculate \( m \cdot R \cdot T \):
\[
W = 1.36 \cdot 0.286 \cdot 350.15 \cdot 2.3026
\]
- Calculate \( m \cdot R \cdot T \):
\[
1.36 \cdot 0.286 = 0.38896 \, \text{kJ/K}
\]
\[
0.38896 \cdot 350.15 \approx 136.286 \, \text{kJ}
\]

- Finally multiply by the logarithm term:
\[
W = 136.286 \cdot 2.3026 \approx 313.385 \, \text{kJ}
\]

Thus, the work done during the isothermal compression of the gas is approximately:

\[
\boxed{313.39 \, \text{kJ}}
\]